Abstract
We consider the Dirichlet and Neumann boundary value problems for the equations
\begin{align}L_u&\equiv\Delta u=v(x,y)\tag{1}\label{1},\L_u&\equiv\Delta u-\lambda u=v(x,y)\tag{2}\label{2}\end{align}
in the square $R[0 \le x, y \le \pi]$. Approximate solutions are sought in the form
\begin{equation}
u_m^{(1)}=\sum_{k=1}^mf_k(x)\sin ky\tag{3}
\label{3}
\end{equation}
for the Dirichlet problem and in the form
\begin{equation}
u_m^{(1)}=\sum_{k=1}^m\varphi_k(x)\cos ky\tag{4}
\label{4}
\end{equation}
for the Neumann problem. According to the method of collocation along lines, the functions $f_k(x)$ and $\varphi_k(x)$ are determined from a system of ordinary differential equations
\begin{gather}Lu_m^{(i)}(x, y_j)=v(x, y_j)\quad(i=1,2)\tag{5}\label{5},\y=y_j\in(0;\pi)\quad(j=1,2,\dots,m;\,i=1),\tag{6}\label{6}\y=y_j\in[0;\pi]\quad(j=0,1,\dots,m;\,i=2)\tag{7}\label{7}\end{gather}
with the boundary conditions
\begin{align}f_k(0)&=f_k(\pi)=0\quad(k=1,2,\dots,m;\,i=1),\tag{8}\label{8}\\varphi'_k(0)&=\varphi'_k(\pi)=0\quad(k=0,1,\dots,m; i=2)\tag{9}\label{9}.\end{align}
Under certain requirements imposed on the function $v(x,y)$ and the choice of collocation lines \eqref{6} and \eqref{7}, we prove the solvability of system \eqref{5} with conditions \eqref{8} or \eqref{9}. Furthermore, we investigate the rate of convergence of the sequences of approximate solutions ${\eqref{3},\eqref{5},\eqref{8}}$ and ${\eqref{4},\eqref{5},\eqref{9}}$ to the corresponding exact solutions $u^{(1)}$ and $u^{(2)}$.
Similar results are obtained for the equation
$$\Delta u-\lambda\sum_{k+l=0}^1a_{kl}\frac{\partial^{k+l}}{\partial x^k\partial y^l}u=v(x,y),\quad a_{kl}=\operatorname{const}$$
with Dirichlet and Neumann boundary conditions specified on the boundary of the square $R[-\pi \le x, y \le \pi]$, as well as for the equation
$$\Delta u-\lambda w(x,y)u=v(x,y)$$
with the boundary conditions
$$u|\gamma=0,\quad\frac{\partial^2u}{\partial x^2}\biggr|\biggr|}=\frac{\partial^2u}{\partial x^2{x=\pi}=\frac{\partial^2u}{\partial y^2}\biggr|=0.$$}=\frac{\partial^2u}{\partial y^2}\biggr|_{y=\pi
Bibliography: 3 items.
Full Text
Preamble
In this section, we consider the boundary value problems for the equation $\Delta u = v(x, y)$ in the domain $D = {0 < x, y < \pi}$. We analyze both the Dirichlet (D) and Neumann (N) conditions on the boundary $\Gamma$. Specifically, for the Neumann problem, we assume the consistency condition $\iint_D v(x, y) dx dy = 0$.
Let the approximate solution be represented in the form:
$$u_m^{(1)}(x, y) = \sum_{k=1}^m f_k(x) \sin ky \quad \text{(for problem D)}$$
$$u_m^{(2)}(x, y) = \sum_{k=0}^m \phi_k(x) \cos ky \quad \text{(for problem N)}$$
The functions $f_k(x)$ and $\phi_k(x)$ are determined by solving the system of ordinary differential equations resulting from the substitution of these forms into the original equation, subject to the boundary conditions $f_k(0) = f_k(\pi) = 0$ and $\phi_k'(0) = \phi_k'(\pi) = 0$.
1. Error Estimates for the Poisson Equation
We define the partial sums of the Fourier series for the source term $v(x, y)$ as $S_m[v(x, y); y]$ for the sine expansion and $C_m[v(x, y); y]$ for the cosine expansion. The approximation error is characterized by the remainders:
$$R_m^{(1)}(v) = \max |v(x, y) - S_m[v(x, y); y]| \to 0 \quad \text{as } m \to \infty$$
$$R_m^{(2)}(v) = \max |v(x, y) - C_m[v(x, y); y]| \to 0 \quad \text{as } m \to \infty$$
The solutions to the boundary value problems can be expressed using the Green's functions $g_D$ and $g_N$:
$$u^{(1)}(x, y) = \iint_D g_D(x, y; \xi, \eta) v(\xi, \eta) d\xi d\eta$$
$$u^{(2)}(x, y) = \iint_D g_N(x, y; \xi, \eta) v(\xi, \eta) d\xi d\eta + C_2$$
By comparing the exact solution $u^{(k)}$ with the approximate solution $u_m^{(k)}$, we establish the following convergence rates for the function and its derivatives:
$$\frac{\partial^{i+j}}{\partial x^i \partial y^j} [u^{(k)} - u_m^{(k)}] = O[R_m^{(k)}(v)] \quad (k=1, 2; \ 0 \le i+j \le 1)$$
$$\max | \Delta [u^{(k)} - u_m^{(k)}] | = O[R_m^{(k)}(v)]$$
2. Extension to the Helmholtz Equation
Consider the equation $\Delta u - \lambda u = v(x, y)$ with Dirichlet or Neumann boundary conditions. We assume $\lambda$ is not an eigenvalue of the corresponding operator. The solution can be represented via an integral equation of the second kind:
$$u(x, y) = \lambda \iint_D g(x, y; \xi, \eta) u(\xi, \eta) d\xi d\eta + \iint_D g(x, y; \xi, \eta) v(\xi, \eta) d\xi d\eta$$
For the Neumann problem, the Green's function $g_N$ is modified to account for the constant term. We define the integral operator $A$ such that the equation becomes $u - \lambda Au = f$. The approximate solution $u_m$ satisfies a corresponding discretized system.
The convergence of the method for the Helmholtz equation is linked to the convergence of the Fourier series of the source term and the kernel. Specifically, if $R_m(k)$ and $R_m(v)$ are the approximation errors for the kernel and the source term respectively, the error in the solution satisfies:
$$\frac{\partial^{i+j}}{\partial x^i \partial y^j} [u - u_m] = O[R_m(k) + R_m(v)] \quad (0 \le i+j \le 3)$$
$$\max | \Delta [u - u_m] | = O[R_m(k) + R_m(v)]$$
3. Generalizations and Higher-Order Operators
The method can be extended to higher-order biharmonic-type equations of the form $\Delta^2 u - \lambda u = v(x, y)$ with appropriate boundary conditions on $D$. Let $u|{y=0, \pi} = 0$ and $\Delta u| = 0$. Using a similar expansion in terms of eigenfunctions $\sin ky$, we reduce the problem to a system of fourth-order ordinary differential equations for the coefficients $f_k(x)$.
The error analysis follows the same logic as the second-order case. If the source term $v(x, y)$ belongs to the class $L_2(D)$ and satisfies the necessary boundary conditions, the approximate solution $u_m$ converges to the exact solution $u$ in the corresponding Sobolev space.
References
- Kantorovich, L. V., and Krylov, V. I. Approximate Methods of Higher Analysis. 1962.
- Smirnov, V. I. A Course of Higher Mathematics, Vol. IV, 1945.
- Nikolsky, S. M. On the solution of boundary value problems by variational methods. Mat. Sbornik, 1963.